Mean Variance Portfolio
Overview
Markowitz's famous paper on portolio optimization hypothesized that investors would seek to maximize the expected return
of a portfolio for a given level of portfolio variance.
Problem Definition
Following
Luenberger,
Minimize the portfolio variance
{% \frac{1}{2} \vec{w}^T \Sigma \vec{w} %}
Subject to the portfolio mean return equal to a given target
{% \vec{w}^T \vec{r} = \bar{r} %}
where
{% \sum w_i = 1 %}
- {% \bar{r} %} - is a target return
- {% \vec{w} %} be
the vector of portfolio weights
- {% r %} is a vector of forecasted asset returns
- {% \Sigma %} is the asset covariance matrix
Solution
The optimal portfolio weights can be solved with a
Langrange multiplier approach, with two Lagrange multipliers
{% \lambda %} and {% \mu %}.
{% \Sigma \vec{w} -\lambda \bar{r} - \mu = 0 %}
{% \vec{w}^T \vec{r} = \bar{r} %}
{% \sum w_i = 1 %}
where
- {% \Sigma %} is the covariance matrix
- {% \vec{w} %} is the weights vector
- {% \bar{r} %} is target return
- {% \vec{r} %} is a vector of forecasted returns
Stated in
matrix
form:
{%
\begin{bmatrix}
\Sigma & \vec{r} & \vec{1} \\
\vec{r}^t & 0 & 0\\
\vec{1}^t & 0 & 0 \\
\end{bmatrix}
\times
\begin{bmatrix}
\vec{w} \\
\lambda \\
\mu \\
\end{bmatrix}
=
\begin{bmatrix}
\vec{0} \\
\bar{r} \\
1 \\
\end{bmatrix}
%}
(see
Luenberger)
Topics
Library
Portfolio Library
